The Binomial Formula

The Binomial Formula, which uses the binomial coefficient ${n \choose k}$, can be used to multiply the expression $(x+y)^n$ out in a convenient way: it expands \( (x+y)^n \) as \[ \sum\limits_{k=0}^n {n \choose k}x^{n-k}y^k.\]

Examples:

• \( (x+y)^2 = {2 \choose 0}x^2 + {2 \choose 1}xy + {2 \choose 2}y^2 \).
• \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \) by applying binomial coefficients.

Binomial coefficients are symmetric because \( {n \choose k} = {n \choose n-k} \).